Linear Regression Prediction Calculator
Calculate linear regression predictions with complete explanations. Learn regression analysis, trend lines, and statistical forecasting. Free calculator.
Quick Answer
Linear regression: y = mx + b where m = slope, b = y-intercept. Finds best-fit line through data points using least squares method. R² measures goodness of fit. Essential for prediction, trend analysis, and forecasting.
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Calculate With Full ToolWhat is Linear Regression?
Linear regression is a statistical method that models the relationship between a dependent variable and one or more independent variables using a linear equation. It finds the best-fit line that minimizes the sum of squared differences between observed and predicted values.
How Linear Regression Works
The method uses the least squares approach to find the line that best represents the data trend. It calculates slope and intercept that minimize the vertical distances between data points and the regression line, providing optimal predictions.
Applications and Uses
Linear regression is widely used for prediction, forecasting, trend analysis, and understanding relationships between variables. Applications include finance, economics, science, engineering, and business analytics for data-driven decision making.
Linear Regression Formulas
Equation: y = mx + b
Slope: m = Σ((x - x̄)(y - ȳ)) / Σ((x - x̄)²)
Intercept: b = ȳ - m(x̄)
Regression Equation: y = mx + b
Slope Formula: m = Σ((x - x̄)(y - ȳ)) / Σ((x - x̄)²)
Intercept Formula: b = ȳ - m(x̄)
Correlation Coefficient: r = Σ((x - x̄)(y - ȳ)) / √(Σ(x - x̄)² × Σ(y - ȳ)²)
Coefficient of Determination: R² = r² (0 ≤ R² ≤ 1)
Step-by-Step Example
Example: Data points (1,2), (2,4), (3,5), (4,4), (5,5)
Step 1: Calculate means: x̄ = 3, ȳ = 4
Step 2: Calculate covariance: Σ((x - x̄)(y - ȳ)) = 6
Step 3: Calculate x-variance: Σ((x - x̄)²) = 10
Step 4: Calculate slope: m = 6 ÷ 10 = 0.6
Step 5: Calculate intercept: b = 4 - 0.6(3) = 2.2
Step 6: Regression equation: y = 0.6x + 2.2
Example: Make prediction for x = 6
Step 1: Use regression equation: y = 0.6x + 2.2
Step 2: Substitute x = 6: y = 0.6(6) + 2.2
Step 3: Calculate: y = 3.6 + 2.2 = 5.8
Step 4: Prediction: When x = 6, y ≈ 5.8
This example shows the complete process of calculating linear regression and making predictions. The regression line y = 0.6x + 2.2 provides the best linear fit for the data points.
Who Should Use This Calculator?
Data Scientists
Build predictive models and analyze trends
Researchers
Analyze relationships between variables
Business Analysts
Forecast sales and market trends
Economists
Model economic relationships and predictions
Frequently Asked Questions
What does R-squared tell us?
R-squared (R²) measures the proportion of variance in the dependent variable that's explained by the independent variable. Values range from 0 to 1, where higher values indicate better model fit and stronger predictive power.
When is linear regression appropriate?
Linear regression works best when there's a linear relationship between variables, residuals are normally distributed, and there's constant variance. It's not suitable for nonlinear relationships without transformation or other methods.
What's the difference between correlation and regression?
Correlation measures the strength and direction of relationship between two variables (symmetric), while regression models how one variable predicts another (asymmetric). Correlation doesn't imply causation, while regression can be used for prediction.
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